Finite size scaling: the connection between lattice QCD

Outline Introduction Fluctuations Three strategies Systematics Summary

Finite size scaling: the connection

between lattice QCD and heavy-ion experiments

Sourendu Gupta

TIFR Mumbai

Quarks and Hadrons under Extreme Conditions, 2011Keio University, Tokyo, Japan

November 18, 2011

SG Phase diagram of QCD


1 Introduction

2 Fluctuations of conserved quantities

3 New approaches to standing questions

4 Systematic errors and intrinsic scales

5 Summary



Outline

1 Introduction




5 Summary



Heavy-ion physics

Experimental observations

Many interesting new phenomena: jet quenching, elliptic flow,strange chemistry, fluctuations of conserved quantities ...



Heavy-ion physics



Systematic understanding

Matter formed: characterized by T and µ. History of fireballdescribed by hydrodynamics and diffusion. Small mean free paths.



Heavy-ion physics



Systematic understanding

Matter formed: characterized by T and µ. History of fireballdescribed by hydrodynamics and diffusion. Small mean free paths.

Theoretical underpinning

Does QCD describe this matter? Is there a new nonperturbativetest of QCD?



Outline

1 Introduction




5 Summary



Fluctuations of conserved quantities

Observations

In a single heavy-ion collision, each conserved quantity (B , Q, S) isexactly constant when the full fireball is observed. In a small partof the fireball they fluctuate: from part to part and event to event.




Observations


Thermodynamics

If ξ3 ≪ Vobs ≪ Vfireball , then fluctuations can be explained in thegrand canonical ensemble: energy and B, Q, S allowed to fluctuatein one part by exchange with rest of fireball (diffusion: transport).




Observations


Thermodynamics

If ξ3 ≪ Vobs ≪ Vfireball , then fluctuations can be explained in thegrand canonical ensemble: energy and B, Q, S allowed to fluctuatein one part by exchange with rest of fireball (diffusion: transport).

Comparison

Is the observed volume small compared to the volume of thefireball? Are observations in agreement with QCDthermodynamics?



Event-to-event fluctuations

)pN∆Net Proton (-20 -10 0 10 20

Num

ber

of E

vent

s

1

10

210

310

410

510

610 0-5%30-40%70-80%

Au+Au 200 GeV<0.8 (GeV/c)

T0.4<p

|y|<0.5

text

STAR

arxiv:1004.4959

Protons are proxy baryonsSG Phase diagram of QCD


Finite size scaling

placeholder

STAR:QM

2009

FSS shape variables change with volume (proxy: Npart)SG Phase diagram of QCD


QCD predictions at finite µB

Madhava-Maclaurin expansion of the (dimensionless) pressure:

1

T 4P(t, z) =

∞∑

n=0

T n−4χ(n)B (t)

zn

n!, (t = T/Tc , z = µ/T )

measure each NLS at µ = 0, sum series expansion to find NLS at

any µ. Shape variables: [Bn] = (VobsT3)T n−4χ

(n)B (t, z). Ratios of

cumulants are state variables:

m1 :[B3]

[B4]=

Tχ(3)B

χ(2)B

= Sσ

m2 :[B4]

[B2]=

Tχ(4)B

χ(2)B

= κσ2

m3 :[B4]

[B3]=

Tχ(4)B

χ(3)B

=κσ

S

SG, 2009



Lattice predictions along the freezeout curve

hadronic quarkyonic

colour superconducting

quark gluon plasma

µ

T

Hadron gas models: Hagedorn, Braun-Munzinger, Stachel, Cleymans, Redlich,

Becattini

Lattice predictions continued from µ = 0 to the freezeout curveSG Phase diagram of QCD



nuclear matter

hadronic quarkyonic


quark gluon plasma

µ

T


Becattini




nuclear matter

hadronic quarkyonic


quark gluon plasma

µ

T


Becattini



Checking the match

4 5 6 10 20 30 100 200-2

-1

0

1

2 (2

)χ/

(4)

χ 2T

2m(B)

0

0.5

1

1.5

2 Exp. Data

(2

)χ/

(3)

χT

Lattice QCD

720 420 210 54 20 10 (MeV)

Bµ

HRG1m

(A)σ

S

2 σ κ

(GeV)NNs

T (MeV)16616516014079

Gavai, SG (2010), STAR (2010), GLMRX (2011)SG Phase diagram of QCD


Outline

1 Introduction




5 Summary



Two earlier suggestions

Gavai, SG (Jan 2010)













The first strategy

Use the chemical freezeout curve and the agreement of data andprediction along it to measure

Tc = 175+1−7 MeV.

GLMRX, 2011



The first strategy: measuring Tc

5 10 20 100 200

/Sσ κ

0

5

10Lattice QCD Exp. Data

=165 MeVcT

=170 MeVcT

=175 MeVcT

=180 MeVcT

=190 MeVcT

(A)

160 170 180 190

2 χ

0

10

20

30 (MeV)-7

+1 =175cT

+ 12χmin

(B)

(GeV)NNs (MeV)cT holder

text

GLMRX,Science

332(2011)1525



The second strategy

Using the Madhava-Maclaurin expansion,

m0 =[B2]

[B]=

χ(2)(t, z)/T 2

χ(1)(t, z)/T 3=

1 +O(

zz∗

)2

z[

1− 3(

zz∗

)]

m3 =[B4]

[B3]=

χ(4)(t, z)

χ(3)(t, z)/T=

1 +O(

zz∗

)2

z[

1− 10(

zz∗

)]

Match lattice predictions and data (including statistical andsystematic errors) assuming knowledge of z∗.



The second strategy: µmetry

0

0.1

0.2

0.3

0.4

From m0 From m3

z =

/T

µ

LO LONLO

NLO

NLO

Pad

e

NLO

Pad

eSG Phase diagram of QCD


A third strategy

Fit m0 and m3 simultaneously to get both z and z∗. Since z∗ is theposition of the critical point: high energy data already givesinformation on the critical point!

Indirect experimental estimate of the critical point

From the highest RHIC energy using both statistical andsystematic errors:

µE

TE≥ 1.7

Compatible with current lattice estimates: but no lattice input.



A third strategy

Fit m0 and m3 simultaneously to get both z and z∗. Since z∗ is theposition of the critical point: high energy data already givesinformation on the critical point!

Indirect experimental estimate of the critical point

From the highest RHIC energy using both statistical andsystematic errors:

µE

TE≥ 1.7

Compatible with current lattice estimates: but no lattice input.

Reduction of systematic errors on m0 and m3 can give estimates ofboth upper and lower limits on the estimate of the critical point.Cross check the BES result by high energy RHIC/LHC data.



Three signs of the critical point

At the critical point ξ → ∞.

1: CLT fails

Scaling [Bn] ≃ V fails: fluctuations remains out of thermalequilibrium. Signals of out-of-equilibrium physics in other signals.

2: Non-monotonic variation

At least some of the cumulant ratios m0, m1, m2 and m3 will notvary monotonically with

√S . If no critical point then m0,3 ∝ 1/z

and m1 ∝ z .

3: Lack of agreement with QCD thermodynamics

Away from the critical point agreement with QCD observed. In thecritical region no agreement.



Outline

1 Introduction




5 Summary



Length scales in thermodynamics

Persistence of memory?

B, Q, S is exactly constant in full fireball volume Vfireball . In a partof the fireball they fluctuate. When Vobs ≪ Vfireball then globalconservation unimportant. Change acceptance to change Vobs .






The central limit theorem

When ξ3 ≪ Vobs , then thermalization possible: by diffusion ofenergy, B, Q, and S to/from Vobs to rest of fireball. Many“fluctuation volumes” implies that thermodynamic fluctuations areGaussian (central limit theorem).






The central limit theorem

When ξ3 ≪ Vobs , then thermalization possible: by diffusion ofenergy, B, Q, and S to/from Vobs to rest of fireball. Many“fluctuation volumes” implies that thermodynamic fluctuations areGaussian (central limit theorem).

Finite size scaling

Since Vobs is finite, departure from Gaussian. Finite size scalingpossible: if equilibrium then relate QCD predictions to finitevolume effects.



Correlation lengths

0.9

0.95

1

1.05

1.1

PS V AV B

µ h/m

h

T=0.95 Tc

Correlation length in thermodynamics defined through staticcorrelator: same as screening lengths. Implies ξ3 ≪ Vobs ; check.Padmanath et al., 2011



Coupling diffusion to flow

Entropy content in B or S small compared to entropy content offull fireball. Coupled relativistic hydro and diffusion equations canthen be simplified to diffusion-advection equation.Which is more important— diffusion or advection? ExaminePeclet’s number

Pe =λv

D=

λv

ξcs= M

λ

ξ.

When Pe ≪ 1 diffusion dominates. When Pe ≫ 1 advectiondominates. Crossover between these regimes when Pe ≃ 1.

Bhalerao and SG, 2009, and in progress



Coupling diffusion to flow

Entropy content in B or S small compared to entropy content offull fireball. Coupled relativistic hydro and diffusion equations canthen be simplified to diffusion-advection equation.Which is more important— diffusion or advection? ExaminePeclet’s number

Pe =λv

D=

λv

ξcs= M

λ

ξ.

When Pe ≪ 1 diffusion dominates. When Pe ≫ 1 advectiondominates. Crossover between these regimes when Pe ≃ 1.

Advective length scale

New length scale: determines when flow overtakes diffusiveevolution—

λ ≃ ξ

M.

Bhalerao and SG, 2009, and in progress



Finite volumes: density sets a scale

When the total number of baryons (baryons + antibaryons)detected is B+, the volume per detected baryon is ζ3 = Vobs/B+.If ζ ≃ ξ then system may not be thermodynamic: controlled whenVobs/ξ

3 → ∞.Events which (by chance) have large B+ take longer to come tochemical equilibrium: important to study these transportproperties. Can one selectively study these rare events?



Finite volumes: density sets a scale

When the total number of baryons (baryons + antibaryons)detected is B+, the volume per detected baryon is ζ3 = Vobs/B+.If ζ ≃ ξ then system may not be thermodynamic: controlled whenVobs/ξ

3 → ∞.Events which (by chance) have large B+ take longer to come tochemical equilibrium: important to study these transportproperties. Can one selectively study these rare events?

On cumulant order

In central Au Au collisions, the measurement of [B6] involvesζ/ξ ≃ 2. Could it be used to study transport? Probe this byseparating out samples with large B+ and studying their statistics.



Protons or baryons?

1 If 1/τ3 is the reaction rate for the slowest process which takesp ↔ n, then the system reaches (isospin) chemical equilibriumat time t ≫ τ3.

2 Once system is at chemical equilibrium, the proton/baryonratio can be expressed in terms of the isospin chemicalpotential: µ3. Since baryons are small component of the netisospin, µ3 can be obtained in terms of the charge chemicalpotential µQ .

3 If not, then is it still possible to extract the shape of the E/Ebaryon distribution?

Asakawa, Kitazawa: 2011



Outline

1 Introduction




5 Summary



Old questions with new answers

1 What is the QCD cross over temperature: Tc? If freezeoutcurve, {T (

√S), µ(

√S)}, is assumed to be known, then Tc

can be found very accurately from the shape of E/Efluctuations of conserved charges.

2 What is the freezeout curve for fluctuations? If Tc is knownwell enough, then the argument can be turned around and thefreezeout curve can be determined from the shape of E/Efluctuations of conserved charges.

3 If the critical point is assumed to lie near the freezeout curve,then its position can be inferred from high energymeasurement without the benefit of lattice predictions, andverified by direct search.

4 Good news for lattice QCD: experimental value of Tc

compatible with known results; critical end point alsocompatible with current experimental results.



Six scales to think of

1 Scale of the persistence of memory, Vfireball . WhenVfireball/Vobs ≫ 1 then overall conservation forgotten.

2 Shortest length scale ξ, controls scale at which diffusion of Bbecomes important.

3 Scale of observation volume, Vobs . Set by the detector.Comparison to lattice works when ξ3 ≪ Vobs ≪ Vfireball .

4 Peclet scale, λ = ξ/M (where M is the Mach number).Controls freeze out of fluctuations.

5 Volume per unit baryon number, ζ3 = Vobs/B+. Events withζ ≃ ξ, may give insight into diffusion time scale.

6 Time scale for reaction p ↔ n, τ3 needs to be understood.



Backup: Is there physics at Tc?

Is the crossover in QCD best thought of as deconfinement or chiralsymmetry restoration?




Is the crossover in QCD best thought of as deconfinement or chiralsymmetry restoration? Maybe neither? Maybe look for influenceon hydrodynamic evolution: slow cooling





Wuppertal-Budapest (2010)





Wuppertal-Budapest (2010)


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Finite size scaling: the connection between lattice QCD